Question

Hundred cards marked with numbers 1 to 100 are placed in a box. If a card is selected randomly from the box, then the probability that the number on the card selected will be a perfect square is (1) \(\frac{1}{100}\) (2) \(\frac{1}{25}\) (3) \(\frac{1}{10}\) (4) \(\frac{9}{10}\)

Short answer

Answer: \(\frac{1}{10}\)

Step-by-step solution

List all perfect squares from 1 to 100

We will list all perfect square numbers that are within the range of 1 to 100. We can begin by showing the square of each integer, starting with 1: 1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64 9² = 81 10² = 100 These are the perfect square numbers within the given range.

Count the perfect square numbers

Now, let's count the perfect square numbers we've listed down in step 1. There are a total of 10 perfect square numbers from 1 to 100.

Calculate the probability

To find the probability of picking a perfect square card, we will divide the number of perfect square numbers by the total number of cards in the box. So, the probability is: $$\frac{10 \,\text{(number of perfect squares)}}{100\,\text{(total number of cards)}}$$ This simplifies to: $$\frac{10}{100} = \frac{1}{10}$$

Match the answer with the given options

The calculated probability is \(\frac{1}{10}\). So, among the given options, the correct answer is option 3. In conclusion, the probability of picking a perfect square card from the box is \(\frac{1}{10}\) (option 3).

Key concepts

Perfect Squares

Perfect squares are integers that are the product of an integer multiplied by itself. In other words, when an integer is squared, the resulting product is a perfect square. This concept is foundational in understanding various mathematical and algebraic concepts, such as factoring and solving quadratic equations.

For instance, in the context of the given exercise, perfect squares between 1 and 100 include the squares of the integers from 1 to 10. Identifying these numbers is crucial as they serve as favorably outcome events when calculating probability in this scenario.

Probability Calculation

Probability calculation is a branch of mathematics that deals with the likelihood of events occurring. It's often expressed as a fraction or a percentage, reflecting the ratio of the favorable outcomes to the total possible outcomes. In our exercise, the method applied to calculate probability is a fundamental one: dividing the count of favorable outcomes by the total number of potential outcomes.

Thus, understanding how probability is calculated paves the way for predictions in a wide range of fields, from games of chance to weather forecasting, and is essential for interpreting results in a meaningful way. Properly grasping this concept is beneficial not just for this problem but for dealing with any probabilistic scenario.

Enumerating Outcomes

Enumerating outcomes is a systematic listing of all possible outcomes of a random experiment or event. It is a critical step in determining the solution to a probability problem because the total number of outcomes is a key denominator in the probability calculation formula.

For this particular exercise, the act of enumerating involves listing all the perfect squares between 1 to 100. This concrete listing helps in visually assessing the proportion of favorable to possible outcomes. Mastering the skill of enumerating outcomes equips students with a powerful tool to tackle more complex probability questions and ensures that none of the potential outcomes are accidentally overlooked in the calculation process.